Polyhedra and surfaces; representation of volumes
Volume questions test whether you can place a solid in space from one datum (an edge, a face in a plane, a diagonal) and obtain its projections, its section by a plane, and sometimes its development. Euler's relation, C + V = A + 2, is the quick check on any polyhedron you draw.
Core rules
- Regular polyhedra (caras/vértices/aristas): tetrahedron 4/4/6, hexahedron 8/6/12, octahedron 6/8/12, dodecahedron 12/20/30, icosahedron 20/12/30.
- Cube of edge a: face diagonal a√2, body diagonal a√3.
- Regular tetrahedron of edge a: height h = a√(2/3) ≈ 0.8165a, with the apex over the circumcentre of the base.
- Octahedron: three equal diagonals mutually perpendicular, each a√2 for edge a.
- Method for a solid on an inclined plane: abat the plane, draw the true shape of the base, desabat it, then raise the height perpendicular to the plane.
- Surfaces: cylinder and cone developments (the cylinder opens into a rectangle of width 2πr; the cone into a circular sector of radius equal to the generatrix g); a plane section of a cone gives ellipse, parabola or hyperbola according to its angle with the axis; the sphere gives a circle in every section.
Worked example
Regular tetrahedron of edge 40 mm resting on PH. The plan is the true-shape equilateral triangle of side 40 mm; its circumradius is 40/√3 = 23.1 mm, and the apex projects onto that circumcentre. Height = 40 × 0.8165 = 32.66 mm, measured perpendicular to PH and drawn in the elevation. Visibility in the elevation: the rear edge is hidden and is drawn dashed.
Traps that cost marks
Using the tetrahedron formula for the cube (or a√3 instead of 0.8165a); putting the apex over the centroid of the drawing rather than the circumcentre; drawing hidden edges with a continuous line; forgetting that the true shape of a section requires a change of plane or an abatimiento; developing a cone with the base radius instead of the generatrix; cutting the cone with a plane through the apex and calling the result a hyperbola.